PLT INSTRUCTOR ON AN INST TRAINING FLT FAILED TO HAVE TRAINEE HOLD SHORT OF RWY WHEN TAXIING OFF LNDG RWY AND XING THE PARALLEL RWY DUE TO MISUNDERSTANDING OF TWR LCL CTLR INSTRUCTIONS FOR LEAVING RWY. AFTER XING AND STOPPING FOR GND CTL TAXI INSTRUCTIONS; CTLR GAVE THEM TAXI TO PARKING INSTRUCTIONS AND REQUESTED THAT THEY CALL THE TWR.

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MADE VISUAL APCH TO RWY 2R AT PDK -- RETURNING FROM IFR TRAINING FLT. DURING ROLLOUT; UNDERSTOOD TWR ATC TO SAY 'MAKE FIRST AVAILABLE TXWY (TXWY D); CROSS RWY 2L; REMAIN ON THIS FREQ.' PIC ACKNOWLEDGED WITH ONLY N-NUMBER. MADE TURNOFF AT TXWY D; TAXIED ACROSS DEP END OF RWY 2L (RWY 2L/20R); STOPPED AT HOLD LINE AT TXWY A AND CONTACTED GND CTL FOR PERMISSION TO TAXI TO PARKING PAD. AT THAT TIME GND GAVE CLRNC FOR TAXIING; AND REQUESTED WE CALL THE TWR BY TELEPHONE. CALLED PDK TWR AND WAS ADVISED THAT WE HAD BEEN INSTRUCTED TO HOLD SHORT OF RWY 2L/20R. NO EXCUSE; BUT THIS IS A PRIME EXAMPLE OF: 1) INCOMPLETE TWR INSTRUCTIONS. 2) TWR CTLR NOT REQUESTING FULL READBACK (ACKNOWLEDGEMENT) FROM PIC. 3) LACK OF COCKPIT MGMNT; IN AS MUCH AS THE CFI WAS HANDLING THE RADIOS AND DID NOT ASSURE PIC HAD FULLY UNDERSTOOD THE TWR XMISSION. 4) AS CFI; I ACKNOWLEDGE THE ISSUE AND SO INSTRUCTED THE PIC (PVTS) IN THE POSTFLT REVIEW. NO CONFLICT OCCURRED. NO OPERATIONAL DEP OR APCH WAS IN PROCESS ON RWY 2L/20R AT THIS TIME.